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Maths-

General

Easy

Question

The range of $k$ for which the inequality $k cos to the power of 2 end exponent invisible function application x minus k cos invisible function application x plus 1 greater or equal than 0 blank for all x element of open parentheses negative infinity comma infinity close parentheses comma$ is

$k < \frac{- 1}{2}$
$\frac{- 1}{2} \leq k \leq 4$
$k > 4$
$\frac{1}{2} \leq k \leq 5$

The correct answer is: $fraction numerator negative 1 over denominator 2 end fraction less or equal than k less or equal than 4$

$k cos to the power of 2 end exponent invisible function application x minus k cos invisible function application x plus 1 greater or equal than 0 for all x element of open parentheses negative infinity comma infinity close parentheses$
$rightwards double arrow k open parentheses cos to the power of 2 end exponent invisible function application x minus cos invisible function application x close parentheses plus 1 greater or equal than 0$
$N o w cos to the power of 2 end exponent invisible function application x minus cos invisible function application x equals open parentheses cos invisible function application x minus fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 2 end exponent minus fraction numerator 1 over denominator 4 end fraction$
$rightwards double arrow negative fraction numerator 1 over denominator 4 end fraction less or equal than cos to the power of 2 end exponent invisible function application x minus cos invisible function application x less or equal than 2$
$W e blank h a v e blank 2 k plus 1 greater or equal than 0 blank a n d minus fraction numerator k over denominator 4 end fraction plus 1 greater or equal than 0$
$H e n c e comma blank minus fraction numerator 1 over denominator 2 end fraction less or equal than k less or equal than 4$

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